Abstract

We present a technique for efficiently computing the reflection and transmission of light by arbitrary systems of turbid layers. To approximate the steady-state reflectance and transmittance without the need to solve difficult boundary conditions, we convolve the reflectance and transmittance profiles of individual layers. We extend single-slab boundary conditions to handle index-of-refraction mismatches between turbid slabs and account for interlayer scattering by applying methods similar to Kubelka–Munk theory in frequency space. We demonstrate good agreement between the reflectance and the transmittance predicted by our model and numerical Monte Carlo methods and show that the far-source reflectance and transmittance of multilayered turbid materials are dominated by interlayer scattering.

© 2006 Optical Society of America

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  1. B. C. Wilson and M. S. Patterson, "The physics of photodynamic therapy," Phys. Med. Biol. 31, 327-360 (1986).
    [CrossRef] [PubMed]
  2. S. L. Jacques and S. A. Prahl, "Modeling optical and thermal distributions in tissue during laser irradiation," Lasers Surg. Med. 6, 494-503 (1987).
    [CrossRef] [PubMed]
  3. M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, "Skin optics," IEEE Trans. Biomed. Eng. 36, 1146-1154 (1989).
    [CrossRef] [PubMed]
  4. A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagnières, and H. van den Bergh, "Noninvasive determination of the optical properties of two-layered turbid media," Appl. Opt. 37, 779-791 (1998).
    [CrossRef]
  5. R. P. Hemenger, "Optical properties of turbid media with specularly reflecting boundaries: applications to biological problems," Appl. Opt. 16, 2007-2012 (1977).
    [CrossRef] [PubMed]
  6. T. J. Farrell, M. S. Patterson, and M. Essenpreis, "Influence of layered tissue architecture on estimates of tissue optical properties obtained from spatially resolved diffuse reflectometry," Appl. Opt. 37, 1958-1972 (1998).
    [CrossRef]
  7. L. V. Wang, S. L. Jacques, and L. Zheng, "MCML—Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
    [CrossRef] [PubMed]
  8. S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, "The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions," Med. Phys. 27, 252-264 (2000).
    [CrossRef] [PubMed]
  9. J. M. Schmitt, G. X. Zhou, and E. C. Walker, "Multilayer model of photon diffusion in skin," J. Opt. Soc. Am. A 7, 2141-2153 (1990).
    [CrossRef] [PubMed]
  10. M. Keijzer, W. M. Star, and P. R. M. Storchi, "Optical diffusion in layered media," Appl. Opt. 27, 1820-1824 (1988).
    [CrossRef] [PubMed]
  11. A. Kienle, T. Glanzmann, G. Wagnières, and H. van den Bergh, "Investigation of two-layered turbid media with time-resolved reflectance," Appl. Opt. 37, 6852-6862 (1998).
    [CrossRef]
  12. J.-M. Tualle, J. Prat, E. Tinet, and S. Avrillier, "Real-space Green's function calculation for the solution of the diffusion equation in stratified turbid media," J. Opt. Soc. Am. A 17, 2046-2055 (2000).
    [CrossRef]
  13. J.-M. Tualle, H. L. Nghiem, D. Ettori, R. Sablong, E. Tinet, and S. Avrillier, "Asymptotic behavior and inverse problem in layered scattering media," J. Opt. Soc. Am. A 21, 24-34 (2004).
    [CrossRef]
  14. I. A. Vitkin, B. C. Wilson, and R. R. Anderson, "Analysis of layered scattering materials by pulsed photothermal radiometry: application to photon propagation in tissue," Appl. Opt. 34, 2973-2982 (1995).
    [CrossRef] [PubMed]
  15. G. Alexandrakis, T. J. Farrell, and M. S. Patterson, "Monte Carlo diffusion hybrid model for photon migration in a two-layer turbid medium in the frequency domain," Appl. Opt. 39, 2235-2244 (2000).
    [CrossRef]
  16. J. Ripoll, V. Ntziachristos, J. P. Culver, D. N. Pattanayak, A. G. Yodh, and M. Nieto-Vesperinas, "Recovery of optical parameters in multiple-layered diffusive media: theory and experiments," J. Opt. Soc. Am. A 18, 821-830 (2001).
    [CrossRef]
  17. F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, "Analytical approximate solutions of the time-domain diffusion equation in layered slabs," J. Opt. Soc. Am. A 19, 71-80 (2002).
    [CrossRef]
  18. S. D. Bianco, F. Martelli, and G. Zaccanti, "Procedure for retrieving the optical properties of a two-layered medium from time-resolved reflectance measurements," Opt. Lett. 36, 4587-4599 (1997).
  19. F. Martelli, A. Sassaroli, S. D. Bianco, Y. Yamada, and G. Zaccanti, "Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method," Phys. Rev. E 67, 056623 (2003).
    [CrossRef]
  20. J. Ripoll and M. Nieto-Vesperinas, "Index mismatch for diffuse photon density waves at both flat and rough diffuse-diffuse interfaces," J. Opt. Soc. Am. A 16, 1947-1957 (1999).
    [CrossRef]
  21. L. V. Wang, S. L. Jacques, and L. Zheng, "CONV—convolution for responses to a finite diameter photon beam incident on multi-layered tissues," Comput. Methods Programs Biomed. 54, 141-150 (1997).
    [CrossRef]
  22. T. J. Farrell and M. S. Patterson, "A diffusion theory model of spatially resolved, steady-state diffuse reflections for the noninvasive determination of tissue optical propertiesin vivo," Med. Phys. 19, 879-888 (1992).
    [CrossRef] [PubMed]
  23. R. A. J. Groenhuis, H. A. Ferwerda, and J. J. T. Bosch, "Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory," Appl. Opt. 22, 2456-2462 (1983).
    [CrossRef] [PubMed]
  24. A. Ishimaru, Wave Propagation and Scattering in Random Media (Oxford U. Press, 1978).
  25. D. Contini, F. Martelli, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory," Appl. Opt. 36, 4587-4599 (1997).
    [CrossRef] [PubMed]
  26. W. G. Egan, T. W. Hilgeman, and J. Reichman, "Determination of absorption and scattering coefficients for nonhomogeneous media. 2: Experiment," Appl. Opt. 12, 1816-1823 (1973).
    [CrossRef] [PubMed]
  27. L. V. Wang, "Rapid modeling of diffuse reflectance of light in turbid slabs," J. Opt. Soc. Am. A 15, 936-944 (1998).
    [CrossRef]
  28. G. W. Faris, "Diffusion equation boundary conditions for the interface between turbid media: a comment," J. Opt. Soc. Am. A 19, 519-520 (2002).
    [CrossRef]
  29. S. Prahl, "Light transport in tissue," Ph.D. thesis (University of Texas at Austin, 1998).
  30. I. Dayan, S. Havlin, and G. H. Weiss, "Photon migration in a two-layer turbid medium: a diffusion analysis," J. Mod. Opt. 39, 1567-1582 (1992).
    [CrossRef]
  31. P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931). English translation by Steve Westin, http:www.graphics.cornell.edu/~westin/pubs/kubelka.pdf.
  32. P. Kubelka, "New contributions to the optics of intensely light-scattering materials. Part II: Non-homogeneous layers," J. Opt. Soc. Am. 44, 330-335 (1954).
    [CrossRef]
  33. M. Birkinshaw, "Radially symmetric Fourier transforms," in Astronomical Data Analysis Software and Systems III, Vol. 61 of ASP Conference Series (Astronomical Society of the Pacific, 1994), pp. 249-252.
  34. S. Eda and E. Okada, "Monte Carlo analysis of near-infrared light propagation in a neonatel head model," Syst. Comput. Japan 35, 60-69 (2004); S. Eda and E. Okada,[Denshi Joho Tsushin Gakkai Ronbunshi J84-D-II, 2654-2661 (2001)].
    [CrossRef]

2004 (2)

J.-M. Tualle, H. L. Nghiem, D. Ettori, R. Sablong, E. Tinet, and S. Avrillier, "Asymptotic behavior and inverse problem in layered scattering media," J. Opt. Soc. Am. A 21, 24-34 (2004).
[CrossRef]

S. Eda and E. Okada, "Monte Carlo analysis of near-infrared light propagation in a neonatel head model," Syst. Comput. Japan 35, 60-69 (2004); S. Eda and E. Okada,[Denshi Joho Tsushin Gakkai Ronbunshi J84-D-II, 2654-2661 (2001)].
[CrossRef]

S. Eda and E. Okada, "Monte Carlo analysis of near-infrared light propagation in a neonatel head model," Syst. Comput. Japan 35, 60-69 (2004); S. Eda and E. Okada,[Denshi Joho Tsushin Gakkai Ronbunshi J84-D-II, 2654-2661 (2001)].
[CrossRef]

2003 (1)

F. Martelli, A. Sassaroli, S. D. Bianco, Y. Yamada, and G. Zaccanti, "Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method," Phys. Rev. E 67, 056623 (2003).
[CrossRef]

2002 (2)

2001 (1)

2000 (3)

1999 (1)

1998 (4)

1997 (3)

1995 (2)

I. A. Vitkin, B. C. Wilson, and R. R. Anderson, "Analysis of layered scattering materials by pulsed photothermal radiometry: application to photon propagation in tissue," Appl. Opt. 34, 2973-2982 (1995).
[CrossRef] [PubMed]

L. V. Wang, S. L. Jacques, and L. Zheng, "MCML—Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

1992 (2)

T. J. Farrell and M. S. Patterson, "A diffusion theory model of spatially resolved, steady-state diffuse reflections for the noninvasive determination of tissue optical propertiesin vivo," Med. Phys. 19, 879-888 (1992).
[CrossRef] [PubMed]

I. Dayan, S. Havlin, and G. H. Weiss, "Photon migration in a two-layer turbid medium: a diffusion analysis," J. Mod. Opt. 39, 1567-1582 (1992).
[CrossRef]

1990 (1)

1989 (1)

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, "Skin optics," IEEE Trans. Biomed. Eng. 36, 1146-1154 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

S. L. Jacques and S. A. Prahl, "Modeling optical and thermal distributions in tissue during laser irradiation," Lasers Surg. Med. 6, 494-503 (1987).
[CrossRef] [PubMed]

1986 (1)

B. C. Wilson and M. S. Patterson, "The physics of photodynamic therapy," Phys. Med. Biol. 31, 327-360 (1986).
[CrossRef] [PubMed]

1983 (1)

1977 (1)

1973 (1)

1954 (1)

1931 (1)

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931). English translation by Steve Westin, http:www.graphics.cornell.edu/~westin/pubs/kubelka.pdf.

Alexandrakis, G.

Anderson, R. R.

Arridge, S. R.

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, "The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions," Med. Phys. 27, 252-264 (2000).
[CrossRef] [PubMed]

Avrillier, S.

Bays, R.

Bianco, S. D.

F. Martelli, A. Sassaroli, S. D. Bianco, Y. Yamada, and G. Zaccanti, "Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method," Phys. Rev. E 67, 056623 (2003).
[CrossRef]

S. D. Bianco, F. Martelli, and G. Zaccanti, "Procedure for retrieving the optical properties of a two-layered medium from time-resolved reflectance measurements," Opt. Lett. 36, 4587-4599 (1997).

Birkinshaw, M.

M. Birkinshaw, "Radially symmetric Fourier transforms," in Astronomical Data Analysis Software and Systems III, Vol. 61 of ASP Conference Series (Astronomical Society of the Pacific, 1994), pp. 249-252.

Bosch, J. J. T.

Contini, D.

Culver, J. P.

Dayan, I.

I. Dayan, S. Havlin, and G. H. Weiss, "Photon migration in a two-layer turbid medium: a diffusion analysis," J. Mod. Opt. 39, 1567-1582 (1992).
[CrossRef]

Dehghani, H.

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, "The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions," Med. Phys. 27, 252-264 (2000).
[CrossRef] [PubMed]

Dögnitz, N.

Eda, S.

S. Eda and E. Okada, "Monte Carlo analysis of near-infrared light propagation in a neonatel head model," Syst. Comput. Japan 35, 60-69 (2004); S. Eda and E. Okada,[Denshi Joho Tsushin Gakkai Ronbunshi J84-D-II, 2654-2661 (2001)].
[CrossRef]

S. Eda and E. Okada, "Monte Carlo analysis of near-infrared light propagation in a neonatel head model," Syst. Comput. Japan 35, 60-69 (2004); S. Eda and E. Okada,[Denshi Joho Tsushin Gakkai Ronbunshi J84-D-II, 2654-2661 (2001)].
[CrossRef]

Egan, W. G.

Essenpreis, M.

Ettori, D.

Faris, G. W.

Farrell, T. J.

Ferwerda, H. A.

Glanzmann, T.

Groenhuis, R. A. J.

Havlin, S.

I. Dayan, S. Havlin, and G. H. Weiss, "Photon migration in a two-layer turbid medium: a diffusion analysis," J. Mod. Opt. 39, 1567-1582 (1992).
[CrossRef]

Hemenger, R. P.

Hilgeman, T. W.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Oxford U. Press, 1978).

Jacques, S. L.

L. V. Wang, S. L. Jacques, and L. Zheng, "CONV—convolution for responses to a finite diameter photon beam incident on multi-layered tissues," Comput. Methods Programs Biomed. 54, 141-150 (1997).
[CrossRef]

L. V. Wang, S. L. Jacques, and L. Zheng, "MCML—Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, "Skin optics," IEEE Trans. Biomed. Eng. 36, 1146-1154 (1989).
[CrossRef] [PubMed]

S. L. Jacques and S. A. Prahl, "Modeling optical and thermal distributions in tissue during laser irradiation," Lasers Surg. Med. 6, 494-503 (1987).
[CrossRef] [PubMed]

Keijzer, M.

Kienle, A.

Kubelka, P.

P. Kubelka, "New contributions to the optics of intensely light-scattering materials. Part II: Non-homogeneous layers," J. Opt. Soc. Am. 44, 330-335 (1954).
[CrossRef]

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931). English translation by Steve Westin, http:www.graphics.cornell.edu/~westin/pubs/kubelka.pdf.

Martelli, F.

Munk, F.

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931). English translation by Steve Westin, http:www.graphics.cornell.edu/~westin/pubs/kubelka.pdf.

Nghiem, H. L.

Nieto-Vesperinas, M.

Ntziachristos, V.

Okada, E.

S. Eda and E. Okada, "Monte Carlo analysis of near-infrared light propagation in a neonatel head model," Syst. Comput. Japan 35, 60-69 (2004); S. Eda and E. Okada,[Denshi Joho Tsushin Gakkai Ronbunshi J84-D-II, 2654-2661 (2001)].
[CrossRef]

S. Eda and E. Okada, "Monte Carlo analysis of near-infrared light propagation in a neonatel head model," Syst. Comput. Japan 35, 60-69 (2004); S. Eda and E. Okada,[Denshi Joho Tsushin Gakkai Ronbunshi J84-D-II, 2654-2661 (2001)].
[CrossRef]

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, "The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions," Med. Phys. 27, 252-264 (2000).
[CrossRef] [PubMed]

Pattanayak, D. N.

Patterson, M. S.

Prahl, S.

S. Prahl, "Light transport in tissue," Ph.D. thesis (University of Texas at Austin, 1998).

Prahl, S. A.

S. L. Jacques and S. A. Prahl, "Modeling optical and thermal distributions in tissue during laser irradiation," Lasers Surg. Med. 6, 494-503 (1987).
[CrossRef] [PubMed]

Prat, J.

Reichman, J.

Ripoll, J.

Sablong, R.

Sassaroli, A.

F. Martelli, A. Sassaroli, S. D. Bianco, Y. Yamada, and G. Zaccanti, "Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method," Phys. Rev. E 67, 056623 (2003).
[CrossRef]

F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, "Analytical approximate solutions of the time-domain diffusion equation in layered slabs," J. Opt. Soc. Am. A 19, 71-80 (2002).
[CrossRef]

Schmitt, J. M.

Schweiger, M.

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, "The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions," Med. Phys. 27, 252-264 (2000).
[CrossRef] [PubMed]

Star, W. M.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, "Skin optics," IEEE Trans. Biomed. Eng. 36, 1146-1154 (1989).
[CrossRef] [PubMed]

M. Keijzer, W. M. Star, and P. R. M. Storchi, "Optical diffusion in layered media," Appl. Opt. 27, 1820-1824 (1988).
[CrossRef] [PubMed]

Sterenborg, H. J. C. M.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, "Skin optics," IEEE Trans. Biomed. Eng. 36, 1146-1154 (1989).
[CrossRef] [PubMed]

Storchi, P. R. M.

Tinet, E.

Tualle, J.-M.

van den Bergh, H.

van Gemert, M. J. C.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, "Skin optics," IEEE Trans. Biomed. Eng. 36, 1146-1154 (1989).
[CrossRef] [PubMed]

Vitkin, I. A.

Wagnières, G.

Walker, E. C.

Wang, L. V.

L. V. Wang, "Rapid modeling of diffuse reflectance of light in turbid slabs," J. Opt. Soc. Am. A 15, 936-944 (1998).
[CrossRef]

L. V. Wang, S. L. Jacques, and L. Zheng, "CONV—convolution for responses to a finite diameter photon beam incident on multi-layered tissues," Comput. Methods Programs Biomed. 54, 141-150 (1997).
[CrossRef]

L. V. Wang, S. L. Jacques, and L. Zheng, "MCML—Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Weiss, G. H.

I. Dayan, S. Havlin, and G. H. Weiss, "Photon migration in a two-layer turbid medium: a diffusion analysis," J. Mod. Opt. 39, 1567-1582 (1992).
[CrossRef]

Wilson, B. C.

Yamada, Y.

F. Martelli, A. Sassaroli, S. D. Bianco, Y. Yamada, and G. Zaccanti, "Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method," Phys. Rev. E 67, 056623 (2003).
[CrossRef]

F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, "Analytical approximate solutions of the time-domain diffusion equation in layered slabs," J. Opt. Soc. Am. A 19, 71-80 (2002).
[CrossRef]

Yodh, A. G.

Zaccanti, G.

Zheng, L.

L. V. Wang, S. L. Jacques, and L. Zheng, "CONV—convolution for responses to a finite diameter photon beam incident on multi-layered tissues," Comput. Methods Programs Biomed. 54, 141-150 (1997).
[CrossRef]

L. V. Wang, S. L. Jacques, and L. Zheng, "MCML—Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Zhou, G. X.

Appl. Opt. (10)

A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagnières, and H. van den Bergh, "Noninvasive determination of the optical properties of two-layered turbid media," Appl. Opt. 37, 779-791 (1998).
[CrossRef]

R. P. Hemenger, "Optical properties of turbid media with specularly reflecting boundaries: applications to biological problems," Appl. Opt. 16, 2007-2012 (1977).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, and M. Essenpreis, "Influence of layered tissue architecture on estimates of tissue optical properties obtained from spatially resolved diffuse reflectometry," Appl. Opt. 37, 1958-1972 (1998).
[CrossRef]

M. Keijzer, W. M. Star, and P. R. M. Storchi, "Optical diffusion in layered media," Appl. Opt. 27, 1820-1824 (1988).
[CrossRef] [PubMed]

A. Kienle, T. Glanzmann, G. Wagnières, and H. van den Bergh, "Investigation of two-layered turbid media with time-resolved reflectance," Appl. Opt. 37, 6852-6862 (1998).
[CrossRef]

I. A. Vitkin, B. C. Wilson, and R. R. Anderson, "Analysis of layered scattering materials by pulsed photothermal radiometry: application to photon propagation in tissue," Appl. Opt. 34, 2973-2982 (1995).
[CrossRef] [PubMed]

G. Alexandrakis, T. J. Farrell, and M. S. Patterson, "Monte Carlo diffusion hybrid model for photon migration in a two-layer turbid medium in the frequency domain," Appl. Opt. 39, 2235-2244 (2000).
[CrossRef]

R. A. J. Groenhuis, H. A. Ferwerda, and J. J. T. Bosch, "Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory," Appl. Opt. 22, 2456-2462 (1983).
[CrossRef] [PubMed]

D. Contini, F. Martelli, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory," Appl. Opt. 36, 4587-4599 (1997).
[CrossRef] [PubMed]

W. G. Egan, T. W. Hilgeman, and J. Reichman, "Determination of absorption and scattering coefficients for nonhomogeneous media. 2: Experiment," Appl. Opt. 12, 1816-1823 (1973).
[CrossRef] [PubMed]

Comput. Methods Programs Biomed. (2)

L. V. Wang, S. L. Jacques, and L. Zheng, "CONV—convolution for responses to a finite diameter photon beam incident on multi-layered tissues," Comput. Methods Programs Biomed. 54, 141-150 (1997).
[CrossRef]

L. V. Wang, S. L. Jacques, and L. Zheng, "MCML—Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Eng. (1)

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, "Skin optics," IEEE Trans. Biomed. Eng. 36, 1146-1154 (1989).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

I. Dayan, S. Havlin, and G. H. Weiss, "Photon migration in a two-layer turbid medium: a diffusion analysis," J. Mod. Opt. 39, 1567-1582 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

Lasers Surg. Med. (1)

S. L. Jacques and S. A. Prahl, "Modeling optical and thermal distributions in tissue during laser irradiation," Lasers Surg. Med. 6, 494-503 (1987).
[CrossRef] [PubMed]

Med. Phys. (2)

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, "The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions," Med. Phys. 27, 252-264 (2000).
[CrossRef] [PubMed]

T. J. Farrell and M. S. Patterson, "A diffusion theory model of spatially resolved, steady-state diffuse reflections for the noninvasive determination of tissue optical propertiesin vivo," Med. Phys. 19, 879-888 (1992).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Med. Biol. (1)

B. C. Wilson and M. S. Patterson, "The physics of photodynamic therapy," Phys. Med. Biol. 31, 327-360 (1986).
[CrossRef] [PubMed]

Phys. Rev. E (1)

F. Martelli, A. Sassaroli, S. D. Bianco, Y. Yamada, and G. Zaccanti, "Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method," Phys. Rev. E 67, 056623 (2003).
[CrossRef]

Syst. Comput. Japan (1)

S. Eda and E. Okada, "Monte Carlo analysis of near-infrared light propagation in a neonatel head model," Syst. Comput. Japan 35, 60-69 (2004); S. Eda and E. Okada,[Denshi Joho Tsushin Gakkai Ronbunshi J84-D-II, 2654-2661 (2001)].
[CrossRef]

Z. Tech. Phys. (Leipzig) (1)

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931). English translation by Steve Westin, http:www.graphics.cornell.edu/~westin/pubs/kubelka.pdf.

Other (3)

M. Birkinshaw, "Radially symmetric Fourier transforms," in Astronomical Data Analysis Software and Systems III, Vol. 61 of ASP Conference Series (Astronomical Society of the Pacific, 1994), pp. 249-252.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Oxford U. Press, 1978).

S. Prahl, "Light transport in tissue," Ph.D. thesis (University of Texas at Austin, 1998).

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Figures (10)

Fig. 1
Fig. 1

(a) Dipole and (b) multipole configuration.

Fig. 2
Fig. 2

Mirroring distance of the multipole is changed in the case of mismatched indices.

Fig. 3
Fig. 3

First two modes of the interscattering of light between two turbid layers.

Fig. 4
Fig. 4

Comparison of the convolved multipole with Monte Carlo simulation using parameters from Fig. 3 of Kienle et al.[4] The parameters are σ s 1 = 1.3 mm 1 , σ s 2 = 1.0 mm 1 , σ a 1 = 0.005 mm 1 , and (a) σ a 2 = 0.01 mm 1 or (b) σ a 2 = 0.022 mm 1 . The thickness of the top layer is d 1 = 6 mm , d 2 = mm , and n 1 = n 2 = 1.4 . KM, Kubelka–Munk.

Fig. 5
Fig. 5

Comparison of the convolved multipole with Monte Carlo simulation using parameters from Fig. 4 of Kienle et al.[4] The parameters of the right plot are σ s 1 = 1.3 mm 1 , σ s 2 = 1.0 mm 1 , σ a 1 = 0.005 mm 1 , and (a) σ a 2 = 0.01 mm 1 or (b) σ a 2 = 0.022 mm 1 . The thickness of the top layer is d 1 = 10 mm , d 2 = mm , and n 1 = n 2 = 1.4 .

Fig. 6
Fig. 6

Relative error of Figs. 4a, 4b.

Fig. 7
Fig. 7

Relative error of Figs. 5a, 5b.

Fig. 8
Fig. 8

(a) Comparison of the convolved multipole with Monte Carlo simulation using parameters from Eda and Okada.[34] The parameters are σ s 1 = 1.882 mm 1 , σ s 2 = 1.584 mm 1 , σ s 3 = 0.452 mm 1 , σ s 4 = 0.963 mm 1 , σ a 1 = 0.018 mm 1 , σ a 2 = 0.016 mm 1 , σ a 3 = 0.048 mm 1 , σ a 2 = 0.037 mm 1 , d 1 = d 2 = 2 mm , d 3 = 3 mm , d 4 = , and n 1 = n 2 = n 3 = n 4 = 1.4 . (b) Relative error with and without the Kubelka–Munk correction to the convolution.

Fig. 9
Fig. 9

Reflectance of a two-layered material (a), without and (b) with the Kubelka–Munk correction. The circles and long-dashed curves are when the material is lit from above, and the crosses and solid curves are when the material is lit from below. The properties of the layers are σ s 1 = 1.0 mm 1 , σ s 2 = 4.0 mm 1 , σ a 1 = 0.005 mm 1 , σ a 2 = 0.001 mm 1 , d 1 = 5 mm , d 2 = 1 mm , n 1 = 1.1 , and n 2 = 1.4 . The order of parameters is reversed to calculate the bottom-lit case. The short-dashed lines show the reflectances of the top and bottom layers individually.

Fig. 10
Fig. 10

Transmittance of a two-layered material (a), without and (b) with the Kubelka–Munk (KM) correction. The circles and dashed curves are when the material is lit from above, and the crosses and solid curves are when the material is lit from below. The parameters of the layers are given in Fig. 9. The total diffuse transmittance in the top-lit case is 8.8% without and 13.0% with the KM correction, compared with 13.1% as calculated by Monte Carlo. In the bottom-lit case, the total diffuse transmittance is 9.1% without and 12.9% with the KM correction, compared with 12.3% as calculated by Monte Carlo. The short-dashed lines show the transmittances of the top and bottom layers individually.

Equations (26)

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Ω + L ( r , s ̂ ) ( z ̂ s ̂ ) d s ̂ = r s ( η ) Ω L ( r , s ̂ ) ( z ̂ s ̂ ) d s ̂ at z = 0 ,
L ( r , s ̂ ) = 1 4 π ϕ ( r ) + 3 4 π E ( r ) s ̂ ,
ϕ ( r ) 2 A D ϕ ( r ) z = 0 ,
ϕ ( r ) = 1 4 π D [ exp ( μ eff r p ) r p exp ( μ eff r n ) r n ] ,
A = 1 + r s ( η ) 1 r s ( η ) ,
r s ( η ) { 0.4399 + 0.7099 η 0.3319 η 2 + 0.0636 η 3 , η < 1 1.4399 η 2 + 0.7099 η + 0.6681 + 0.0636 η , η > 1 } .
R ( r ) = D ϕ ( r ) ,
R ( r ) = α 4 π [ z p ( 1 + r p μ eff ) exp ( r p μ eff ) r p 3 z n ( 1 + r n μ eff ) exp ( r n μ eff ) r n 3 ] ,
R d = 0 R ( r ) 2 π r d r .
Ω L ( r , s ̂ ) ( z ̂ s ̂ ) d s ̂ = r s ( η ) Ω + L ( r , s ̂ ) ( z ̂ s ̂ ) d s ̂ at z = d ,
ϕ ( r ) + 2 A D ϕ ( r ) z = 0 at z = d .
z p , i = 2 i ( d + 2 z b ) + z 0 ,
z n , i = 2 i ( d + 2 z b ) z 0 2 z b ,
ϕ ( r ) = i = n n 1 4 π D [ exp ( r p , i μ eff ) r p , i exp ( r n , i μ eff ) r n , i ] ,
R ( r ) = i = n n α z p , i ( 1 + r p , i μ eff ) e r p , i μ eff 4 π r p , i 3 α z n , i ( 1 + r n , i μ eff ) e r n , i μ eff 4 π r n , i 3 ,
T ( r ) = i = n n α ( d z p , i ) ( 1 + r p , i μ eff ) e i r p , i μ eff 4 π r p , i 3 α ( d z n , i ) ( 1 + r n , i μ eff ) e r n , i μ eff 4 π r n , i 3 .
ϕ ( r ) 2 A top D ϕ ( r ) z = 0 at z = 0 ,
ϕ ( r ) + 2 A bottom D ϕ ( r ) z = 0 at z = d ,
A top = 1 + r s ( η top ) 1 r s ( η top ) , A bottom = 1 + r s ( η bottom ) 1 r s ( η bottom ) .
z p , i = 2 i ( d + z b , top + z b , bottom ) + z 0 ,
z n , i = 2 i ( d + z b , top + z b , bottom ) z 0 2 z b , top ,
T 12 ( r ) = T 1 ( r ) T 2 ( r ) .
T 12 = T 1 T 2 + T 1 R 2 R 1 T 2 + T 1 R 2 R 1 R 2 R 1 T 2 + ,
T 12 = T 1 T 2 + T 1 R 2 R 1 T 2 + T 1 R 2 R 1 R 2 R 1 T 2 + = T 1 T 2 [ 1 + R 2 R 1 + ( R 2 R 1 ) 2 + ( R 2 R 1 ) 3 + ] ,
T 12 = T 1 T 2 1 R 2 R 1 .
R 12 = R 1 + T 1 R 2 T 1 1 R 2 R 1 ,

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